No fear! This will get MUCH simpler with the new redesign which focuses a lot on this. It will be rolled out on a limited basis starting this week, rolled out to the entire platform in the following weeks.
In the meantime, here are some answers to your questions.
1 - Set your Pickaxe to unlisted. It will be usable to anyone with the link, but not indexed in our search.
2 - Turn on the cloneable toggle. This will allow you to clone the Pickaxe.
3 - Pickaxe privacy settings dont matter once you put them in a studio. Anything in a studio is overwritten by the Studio privacy settings.
4 - If you clone a studio, you DO NOT clone the users in it.
5 - Just embed the studio! You can then set an embedded usage limit that only applies to embedded instances of the Pickaxe.